Surface area of a rectangular pyramid

0:00- [Instructor] We're told the height of the following right rectangular pyramid is 11 meters. What is the lateral surface area of the pyramid and what is the total surface area of the pyramid? Have a go at this and then we'll do this together. Okay, so first of all, lateral surface area is going around the sides of this three-dimensional figure, what that total surface area is.

0:20If you take that surface area, the lateral surface area, and then add it to the base right over here, the surface area of the base, then you're gonna get the total surface area. So let's just start with lateral and then we can figure out total after that. Well, to figure out lateral, we have to figure out the surface area or the area of each of these four triangular surfaces.

0:42And this being a right rectangular pyramid, we can assume that its peak, its top right over there is right over the center of the base. Now, why does that help us? Well, this is gonna help us recognize that this side over here in the back has the same surface area as this side here on the right, I guess this is on the left, this is on the right, and the side in the back, that over there, is gonna have the same surface area as the side on the front right over there.

1:13What we also know is that since this is at the center, we know that this distance is gonna be half of this 10, so it's going to be five meters, and we know that this distance right over here is going to be half of this six, is going to be three meters. So the first thing we can do to figure out the height of this right-hand side, which is going to the same as the height of the left-hand side, is we can use the Pythagorean theorem.

1:38So how do we do that? Well, we know that 11 squared plus three squared is going to be equal to this height squared. Let's call this h. So we say h squared is equal to 11 squared, trying to find that same color, 11 squared plus three squared. And we're assuming everything is in meters.

1:58Three squared. And so we get h is going to be equal to the square root of 121. 121. Plus nine. Plus nine. Which is equal to the square root of 130. Square root of 130. And let's just leave it like that, and let's just assume that you can use a calculator in this and we need to round to the nearest square meter when we're all done.

2:27So it might be simple just to leave this and not try to simplify that. So that's the height of this side. And now if we wanna figure out its area, we know an area of a triangle is just 1/2 times base times height. So this area, let me do this in a color that you can see, this area right over here is going to be equal to 1/2, our base is 10, times our height, which is square root of 130.

2:55So this is going to be equal to 1/2 times 10 is five. So it's gonna be five times the square root of 130. Now, there's two surfaces with that area, the one on the right and this one that you can't see, or you can only see if this is transparent, on the left. So our lateral surface here, we're gonna have two of those.

3:13Five square roots of 130, plus another five square roots of 130. Actually, let me just write it out so you can see that I have four areas. And now let's do the front and the back. Well, what is going to be the height of this front? Well, we can use the Pythagorean theorem again to figure this out.

3:35And I don't know, let me just call that x since I already used h. Well, we know that x squared, the hypotenuse squared is going to be equal to this length, five squared, and we can just say that's going to be 25, plus this length, or this height I guess I could say, 11 squared.

3:54So plus 121. Plus 121. And what's 25 plus 121? That is, let's see, that's 146. And so we get x is equal to 146, x is equal to, or I should say is equal to the square root of 146. And so if I want to figure out the area of this triangle that's out there in the front, I just multiply 1/2 times the base, times six, times the height, times square root of 146, which is three times the square root of 146.

4:35So let me add that to the lateral surface area, plus three times the square root of 146. And then this thing in the back here, that area is gonna be the same thing. So plus three times the square root of 146. You add it all together, we are going to get, you add these top two, you're going to get 10 times the square root of 130.

4:54You add these bottom two, plus 6 times the square root of 146, and we have our lateral surface area. If you wanted to have that and round it to the nearest square meter, so it's going to be in meters squared, if I were to put parentheses around it like that. And then if I wanted to figure out the total surface area, I would start with the lateral surface area.

5:16Let me start with that. And I would add the surface area of the base. And that one's pretty straightforward, the width times the depth or the width times, yeah, call this width times depth. Six times 10 is going to be 60 square meters. So plus 60 square meters.

5:32So this top one is lateral surface area and then this one that includes the area of the base is total surface area.